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refactor: Update supercommute notation

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jstoobysmith 2025-02-13 10:44:15 +00:00
parent d2ce55ddd0
commit 82fae67ba3
12 changed files with 197 additions and 188 deletions
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22
HepLean/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
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@ -239,7 +239,7 @@ lemma normalOrderF_swap_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * [ofCrAnOpF φc, ofCrAnOpF φa]ₛca * b) = 0 := by
𝓝ᶠ(a * [ofCrAnOpF φc, ofCrAnOpF φa]ₛF * b) = 0 := by
simp only [superCommuteF_ofCrAnOpF_ofCrAnOpF, instCommGroup.eq_1, Algebra.smul_mul_assoc]
rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc, ← mul_assoc, ← mul_assoc,
normalOrderF_swap_create_annihilate φc φa hφc hφa]
@ -248,7 +248,7 @@ lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnFieldOp)
(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φc]ₛca * b) = 0 := by
𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φc]ₛF * b) = 0 := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF_symm]
simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
@ -402,9 +402,9 @@ TODO "Split the following two lemmas up into smaller parts."
lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
(φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnFieldOp) :
(𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φc, ofCrAnOpF φc']ₛca * ofCrAnListF φs')) =
(𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φc, ofCrAnOpF φc']ₛF * ofCrAnListF φs')) =
normalOrderSign (φs ++ φc' :: φc :: φs') •
(ofCrAnListF (createFilter φs) * [ofCrAnOpF φc, ofCrAnOpF φc']ₛca *
(ofCrAnListF (createFilter φs) * [ofCrAnOpF φc, ofCrAnOpF φc']ₛF *
ofCrAnListF (createFilter φs') * ofCrAnListF (annihilateFilter (φs ++ φs'))) := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, mul_sub, sub_mul, map_sub]
conv_lhs =>
@ -463,10 +463,10 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
(φs φs' : List 𝓕.CrAnFieldOp) :
𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * ofCrAnListF φs') =
𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF * ofCrAnListF φs') =
normalOrderSign (φs ++ φa' :: φa :: φs') •
(ofCrAnListF (createFilter (φs ++ φs'))
* ofCrAnListF (annihilateFilter φs) * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca
* ofCrAnListF (annihilateFilter φs) * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF
* ofCrAnListF (annihilateFilter φs')) := by
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, mul_sub, sub_mul, map_sub]
conv_lhs =>
@ -533,7 +533,7 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
-/
lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
[ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛca =
[ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛF =
ofCrAnListF φs * 𝓝ᶠ(ofCrAnListF φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs := by
simp only [normalOrderF_ofCrAnListF, map_smul, superCommuteF_ofCrAnListF_ofCrAnListF,
@ -541,7 +541,7 @@ lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.C
Algebra.mul_smul_comm, mul_comm, Algebra.smul_mul_assoc]
lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca =
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛF =
ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs := by
rw [ofFieldOpListF_sum, map_sum, Finset.mul_sum, Finset.smul_sum, Finset.sum_mul,
@ -561,20 +561,20 @@ lemma ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φs : List
(φs' : List 𝓕.FieldOp) :
ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') =
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs
+ [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
+ [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by
simp [ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF]
lemma ofCrAnOpF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : ofCrAnOpF φ * 𝓝ᶠ(ofFieldOpListF φs') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnOpF φ
+ [ofCrAnOpF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
+ [ofCrAnOpF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by
simp [← ofCrAnListF_singleton, ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF]
lemma anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp)
(φs' : List 𝓕.FieldOp) :
anPartF φ * 𝓝ᶠ(ofFieldOpListF φs') =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs' * anPartF φ)
+ [anPartF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
+ [anPartF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by
rw [normalOrderF_mul_anPartF]
match φ with
| .inAsymp φ => simp